Let $\mathbf{a} = \begin{pmatrix} -3 \\ 10 \\ 1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 5 \\ \pi \\ 0 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -2 \\ 7 \end{pmatrix}.$  Compute
\[(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})].\]
Solution: Expanding $(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a}),$ we get
\begin{align*}
(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a}) &= \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} - \mathbf{c} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \\
&= \mathbf{b} \times \mathbf{c} + \mathbf{a} \times \mathbf{b} - \mathbf{0} + \mathbf{c} \times \mathbf{a} \\
&= \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \\
\end{align*}Then
\begin{align*}
(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})] &= (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}) \\
&= \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) + \mathbf{a} \cdot (\mathbf{c} \times \mathbf{a}) \\
&\quad - \mathbf{b} \cdot (\mathbf{a} \times \mathbf{b}) - \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) - \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}).
\end{align*}Since $\mathbf{a} \times \mathbf{b}$ is orthogonal to $\mathbf{a},$ $\mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = 0.$  Similarly, other dot products vanish, and we are left with
\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) - \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}).\]From the scalar triple product, $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}),$ so this becomes $\boxed{0}.$